Currently, each message gets its own pop-up. Old pop-ups are pushed upward on the screen by newer pop-ups.

If someone writes a bunch of short messages in a row (as people tend to do with instant messaging), the right side of the screen can quickly become filled with pop-ups. This prevents them from seeing or clicking on site content until they manually close each popup.

Here are some examples of how it looks on my desktop and mobile layouts:
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If each message from a given user -replaced- the previous pop-up from that user (instead of pushing it up the screen), the problem would be resolved very tidily, I think.